Vlad #3

To finish up Tuesday, as we now approach the end of Friday, I present Vlad Dokchitser's third lecture.

Compatibility of BSD w/ isogenies

Lemma: E/K, phi:E->E' an isogeny over K

then

(i) L(E/K,s)=L(E'/K,s) (in that the conjectural formulas are the same and where defined they are equal)

(ii) rk E/K = rk E'/K

Proof: the l-adic representations are equal, and isogenies must preserve rank due to having finite kernel and cokernel (since the dual isogeny gives you a multiplication by degree map).

Write the big conjectural formula from BSD as BSD(E/K). Then there is a theorem due to Cassels (extended by Tate to algebraic varieties): if phi is an isogeny from E to E' over K and Sha(E/K) finite, then Sha(E'/K) finite and BSD(E/K)=BSD(E'/K).

Corollary (due to Birch): if phi is an isogeny of prime degree p, then rk(E/K) is ord_p of the product of Cv and Omega_v of (E,w_E) over the product of Cv and Omega_v of (E',w_E').

Proving this essentially uses the definition of the regulator on the equality of the BSD parts.

Now suppose we have two families of elliptic curves over families of fields, Ei/Ki and E'j/K'j, such that the product of the L functions of each family is equal.

Then the sums of the ranks of each family is equal, the finiteness of Sha for one family implies it for the other family, and the products of the BSD(E/K) over each family are equal.